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Generalized Three-Step Numerical Methods for Solving Equations in Banach Spaces

Author

Listed:
  • Michael I. Argyros

    (Department of Computer Science, University of Oklahoma, Norman, OK 73019, USA)

  • Ioannis K. Argyros

    (Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA)

  • Samundra Regmi

    (Department of Mathematics, University of Houston, Houston, TX 77204, USA)

  • Santhosh George

    (Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Mangaluru 575 025, India)

Abstract

In this article, we propose a new methodology to construct and study generalized three-step numerical methods for solving nonlinear equations in Banach spaces. These methods are very general and include other methods already in the literature as special cases. The convergence analysis of the specialized methods is been given by assuming the existence of high-order derivatives which are not shown in these methods. Therefore, these constraints limit the applicability of the methods to equations involving operators that are sufficiently many times differentiable although the methods may converge. Moreover, the convergence is shown under a different set of conditions. Motivated by the optimization considerations and the above concerns, we present a unified convergence analysis for the generalized numerical methods relying on conditions involving only the operators appearing in the method. This is the novelty of the article. Special cases and examples are presented to conclude this article.

Suggested Citation

  • Michael I. Argyros & Ioannis K. Argyros & Samundra Regmi & Santhosh George, 2022. "Generalized Three-Step Numerical Methods for Solving Equations in Banach Spaces," Mathematics, MDPI, vol. 10(15), pages 1-28, July.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:15:p:2621-:d:872863
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    References listed on IDEAS

    as
    1. Zhanlav, T. & Otgondorj, Kh., 2021. "Higher order Jarratt-like iterations for solving systems of nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 395(C).
    2. Xiao, Xiaoyong & Yin, Hongwei, 2017. "Achieving higher order of convergence for solving systems of nonlinear equations," Applied Mathematics and Computation, Elsevier, vol. 311(C), pages 251-261.
    3. Ioannis K. Argyros, 2021. "Unified Convergence Criteria for Iterative Banach Space Valued Methods with Applications," Mathematics, MDPI, vol. 9(16), pages 1-15, August.
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